Let $\Omega \subseteq \mathbb{R}^n$ be a domain. Suppose $u$ is locally integrable (i.e. $u\in L_{loc}^1(\Omega)$) and has a locally integrable weak derivative $\partial_i u$.
Is there a way to find the weak derivative of $\lvert u \rvert$?
I tried to show that $\partial_i \lvert u \rvert = \chi_{u>0}\partial_i u - \chi_{u<0}\partial_i u$. For this, I picked an arbitrary text function $\varphi$ and (using the dominated convergence theorem) showed that $$ \int_{\Omega} \left(\chi_{u>0}\partial_i u - \chi_{u<0}\partial_i u\right)\phi = \lim_{\epsilon \searrow 0} \int_{\Omega} \left[\left(\chi_{u>0} - \chi_{u<0}\right)\phi\right]_\epsilon\cdot \partial_i u $$ where $\left[\left(\chi_{u>0} - \chi_{u<0}\right)\phi\right]_\epsilon$ denotes the mollified version of $\left(\chi_{u>0} - \chi_{u<0}\right)\phi$.
Then, since $\left[\left(\chi_{u>0} - \chi_{u<0}\right)\phi\right]_\epsilon$ is also a test function, there holds $$ \int_{\Omega} \left[\left(\chi_{u>0} - \chi_{u<0}\right)\phi\right]_\epsilon\cdot \partial_i u = -\int_{\Omega} \partial_i\left[\left(\chi_{u>0} - \chi_{u<0}\right)\phi\right]_\epsilon\cdot u $$
Unfortunetly, this doesn't seem to be going anywhere? Any input is appreciated!